Math 9G Geometry Flashcards

Geometric Inequalities

Concepts of Inequality in Segments and Angles

• Definitions of Geometric Inequalities:

  • Segment Inequality: $\overline{AB} < \overline{CD}$ if and only if $AB < CD$.

  • Angle Inequality: $\angle A < \angle B$ if and only if $m\angle A < m\angle B$.

• Fundamental Properties of Inequality:

  • Trichotomy Property: For every real number $x$ and $y$, EXACTLY ONE of the following conditions holds:     $x = y$     $x < y$     $x > y$

  • Transitive Property:

  • If $x < y$ and $y < z$, then $x < z$.

  • If $x > y$ and $y > z$, then $x > y$ .

  • Addition Property:

  • If $a < b$ and $x \le y$, then $a + x < b + y$.

  • Multiplication Property:

  • If $x < y$ and $a > 0$, then $a \cdot x < a \cdot y$.

• Property Identification Exercises:

  1. If $a > 5$ and $b < 5$, then $b < a$ illustrates the Transitive Property.

  2. If $4 < 6$ and $10 < 15$, then $14 < 21$ illustrates the Addition Property.

  3. If $m\angle A < 45$, then $m\angle A \neq 45$ illustrates the Trichotomy Property.

  4. If $a - b = 7$ and $b < 5$, then $a < 12$ illustrates the Addition Property.

  5. If $3a < 2$ and $b > 0$, then $3ab < 2b$ illustrates the Multiplication Property.

• Theorem 7–1:

  • Statement: If $a = b + c$ and $c > 0$, then $a > b$.

  • Two-Column Proof:

• Applications of Theorem 7–1:

  • Example 1: If $G, H, K$ are three points such that $G - H - K$, explain why $GH < GK$.

• Example 2: In $\Delta ACD$, $AC = DC$ and $AD = DB$. Prove that $\angle ABD > \angle ADB$.

The Exterior Angle Theorem and Its Corollaries

• Exterior Angle and Remote Interior Angles:

  • Exterior Angle Definition: If $C$ is between $A$ and $D$, then $\angle BCD$ is an exterior angle of $\Delta ABC$.

  • Remote Interior Angles Definition: $\angle A$ and $\angle B$ of $\Delta ABC$ are called the remote interior angles of the exterior angles $\angle BCD$ and $\angle ACE$.

• Theorem 7–2 (The Exterior Angle Theorem):

  • Statement: An exterior angle of a triangle is greater than each of its remote interior angles.

  • Restatement: Given $\Delta ABC$. If $C$ is between $A$ and $D$, then $\angle BCD > \angle A$ and $\angle BCD > \angle B$.

• Two-Column Proof:

• Corollary 7–2.1:

  • Statement: If a triangle has one right angle, then its other angles are acute.

  • Restatement: Given $\Delta ABC$ with right angle $\angle BCA$ ($m\angle BCA = 90$). Its other angles, $\angle A$ and $\angle B$, have measure less than $90$

  • Two-Column Proof:

Congruence Theorems and Single Triangle Inequalities

• SAA Correspondence Definition: Given a correspondence $ABC \leftrightarrow DEF$ between two triangles: If a pair of corresponding sides are congruent, and two pairs of corresponding angles are congruent, then the correspondence is called an SAA Correspondence.

• Theorem 7–3 (The SAA Theorem):

  • Statement: Every SAA correspondence is a congruence.

  • Proof Outline: Uses Trichotomy Property. It can be shown that $AB \not> DE$ and $AB \not< DE$. Thus, $AB = DE$ is the only remaining possibility, making the triangles congruent by the ASA Postulate.

• Theorem 7–4 (The Hypotenuse-Leg Theorem):

  • Statement: Given a correspondence between two triangles. If the hypotenuse and one leg of one of the triangles are congruent to the corresponding parts of the second triangle, then the correspondence is a congruence.

  • Restatement: Given $\Delta ABC$ and $\Delta DEF$, where $\angle BAC$ and $\angle EDF$ are right angles; $\overline{BC} \cong \overline{EF}$ (hypotenuse) and $\overline{AB} \cong \overline{DE}$ (leg), then $\Delta ABC \cong \Delta DEF$

  • Two-Column Proof:

• Theorem 7–5 (Side-Angle Inequality in a Single Triangle):

  • Statement: If two sides of a triangle are not congruent, then the angles opposite them are not congruent, and the larger angle is opposite the longer side.

  • Restatement: Given $\Delta ABC$ where $\overline{AC} \not\cong \overline{AB}$, if $AC > AB$, then $m\angle B > m\angle C$.

  • Two-Column Proof:

• Theorem 7–6 (Angle-Side Inequality in a Single Triangle):

  • Statement: If angles of a triangle are not congruent, then the sides opposite them are not congruent, and the longer side is opposite the larger angle.

  • Restatement: Given $\Delta ABC$ where $m\angle B \not\cong m\angle C$, if $m\angle B > m\angle C$, then $AC > AB$.

  • Two-Column Proof:

• Theorem 7–7 (The First Minimum Theorem):

  • Statement: The shortest segment joining a point to a line is the perpendicular segment.

  • Restatement: Given line $\ell$ and an external point $P$. If $\overline{PQ} \perp \ell$ at $Q$, and $R$ is any other point of $\ell$, then $PQ < PR$.

  • Two-Column Proof:

• Definition of Distance:

  • The distance between a line and an external point is the length of the perpendicular segment from the point to the line.

  • The distance between a line and a point on the line is defined to be zero.

• Theorem 7–8 (The Triangle Inequality Theorem):

  • Statement: The sum of the lengths of any two sides of a triangle is greater than the length of the third side.

  • Restatement: In any $\Delta ABC$, $AB + BC > AC$, $AB + AC > BC$, or $AC + BC > AB$.

  • Two-Column Proof (for $AB + BC > AC$):

The Hinge Theorem and Converse Hinge Theorem

• Theorem 7–9 (The Hinge Theorem):

  • Statement: If two sides of one triangle are congruent, respectively, to two sides of a second triangle, and the included angle of the first triangle is larger than the included angle of the second, then the third side of the first triangle is longer than the third side of the second.

  • Restatement: Given $\Delta ABC$ and $\Delta DEF$, with $\overline{AB} \cong \overline{DE}$ and $\overline{AC} \cong \overline{DF}$. If $\angle A > \angle D$, then $\overline{BC} > \overline{EF}$.

  • Two-Column Proof:

• Sample Applications:

  • Algebraic Problem: Given $\Delta ABD$ and $\Delta CDB$ sharing side $\overline{BD}$, with $AB = CD$ and $\angle ABD = 44^\circ > \angle CDB = 35^\circ$.

* By Hinge Theorem: $AD > BC \implies 2x + 12 > 4x - 8 \implies 20 > 2x \implies x < 10$

• Real-World Navigation Problem: Given $\Delta ABC$ and $\Delta ABD$ sharing $\overline{AB}$, where $BC = BD = 181\,miles$, $AB = 100\,miles$, $\angle ABC = 48^\circ$, and $\angle ABD = 113^\circ$.

* Since $\angle ABD > \angle ABC$, by the Hinge Theorem, $AD > AC$. Therefore, Plane 2 is closer to the airport than Plane 1.

• Theorem 7–10 (The Converse Hinge Theorem):

  • Statement: If two sides of one triangle are congruent, respectively, to two sides of a second triangle, and the third side of the first triangle is longer than the third side of the second, then the included angle of the first triangle is larger than the included angle of the second.

  • Restatement: Given $\Delta ABC$ and $\Delta DEF$, with $AB = DE$ and $AC = DF$. If $BC > EF$, then $\angle A > \angle D$.

  • Proof Outline: By Trichotomy Property, the possible cases for $\angle A$ and $\angle D$ are $\angle A \cong \angle D$, $\angle A < \angle D$, or $\angle A > \angle D$. Assuming $\angle A \cong \angle D$ yields $BC = EF$ (contradiction). Assuming $\angle A < \angle D$ yields $BC < EF$ by Hinge Theorem (contradiction). Thus, $\angle A > \angle D$ must hold.

Parallelism

Lines in Space and Transversal Angle Pairs

• Relative Positions of Two Lines in Space:

  • Intersecting Lines: Lines that meet at exactly one point. By Theorem 3–4, two intersecting lines determine exactly one plane (they are coplanar).

  • Parallel Lines: Lines that lie in the same plane and do not intersect.

  • Skew Lines: Lines that do not intersect and do not lie in the same plane (non-coplanar).

• Fundamental Theorems on Parallel Lines:

  • Theorem 9–1: Two parallel lines lie in exactly one plane.

  • Theorem 9–2: In a plane, two lines are parallel if they are both perpendicular to the same line.

  • Theorem 9–3 (Existence of Parallels): Let $L$ be a line and $P$ be a point NOT on $L$. Then there is at least one line through $P$, parallel to $L$

• Transversals and Associated Angle Pairs:

  • Transversal Definition: A transversal of two coplanar lines is a line that intersects them in two distinct points.

  • Alternate Interior Angles: Given two lines cut by a transversal $T$ at $P$ and $Q$, angles $\angle APQ$ and $\angle BQP$ on opposite sides of $T$ between the lines are alternate interior angles.

  • Corresponding Angles: An interior angle and an exterior angle on the same side of the transversal that are in corresponding relative positions.

• Congruence Theorems for Transversals:

  • Theorem 9–4: If two lines are cut by a transversal, and one pair of alternate interior angles are congruent, then the other pair of alternate interior angles are also congruent.

  • Theorem 9–6: Given two lines cut by a transversal. If a pair of corresponding angles are congruent, then a pair of alternate interior angles are congruent.

Conditions Guaranteeing Parallelism

• Theorem 9–5 (The AIP Theorem - Alternate Interior Angles Theorem):

  • Statement: Given two lines cut by a transversal, if a pair of alternate interior angles are congruent, then the lines are parallel.

  • Restatement: If $\angle y \cong \angle x$, then $\ell_1 \parallel \ell_2$.

• Theorem 9–7 (The CAP Theorem - Corresponding Angles Theorem):

  • Statement: Given two lines cut by a transversal, if a pair of corresponding angles are congruent, then the lines are parallel.

  • Restatement: If $\angle z \cong \angle x$, then $\ell_1 \parallel \ell_2$.

• Interior Angles on the Same Side of Transversal:

  • Definition: If $\angle x$ and $\angle y$ are alternate interior angles, and $\angle v$ and $\angle x$ form a linear pair, then $\angle x$ and $\angle w$ are interior angles on the same side of the transversal.

• Theorem 9–8:

  • Statement: Given two lines cut by a transversal, if a pair of interior angles on the same side of the transversal are supplementary, the lines are parallel.

  • Restatement: If $\angle v$ and $\angle x$ are supplementary, then $\ell_1 \parallel \ell_2$.

  • Two-Column Proof:

The Parallel Postulate and Properties of Parallel Lines

• Postulate 18 (The Parallel Postulate):

  • Through a given external point there is ONLY ONE line parallel to a given line.

• Properties Derived from Parallel Lines:

  • Theorem 9–9 (The PAI Theorem - Parallel Alternate Interior Theorem): If two parallel lines are cut by a transversal, then alternate interior angles are congruent ($\ell_1 \parallel \ell_2 \implies \angle 1 \cong \angle 2$).

  • Corollary 9–9.1 (The PCA Corollary - Parallel Corresponding Angles Corollary): If two parallel lines are cut by a transversal, each pair of corresponding angles are congruent ($\ell_1 \parallel \ell_2 \implies \angle 2 \cong \angle 3$).

  • Corollary 9–9.2: If two parallel lines are cut by a transversal, the interior angles on the same side of the transversal are supplementary ($\ell_1 \parallel \ell_2 \implies m\angle 2 + m\angle 4 = 180$).

• Transitivity and Intersection Theorems:

  • Theorem 9–10: In a plane, if a line intersects one of two parallel lines in only one point, then it intersects the other.

  • Theorem 9–11: In a plane, if two lines are each parallel to a third line, then they are parallel to each other ($\ell_1 \parallel \ell_3$ and $\ell_2 \parallel \ell_3 \implies \ell_1 \parallel \ell_2$).

  • Theorem 9–12: In a plane, if a line is perpendicular to one of two parallel lines, it is perpendicular to the other ($\ell_1 \parallel \ell_2$ and $\ell_3 \perp \ell_1 \implies \ell_3 \perp \ell_2$).

Measures of Angles in a Triangle

• Theorem 9–13 (Triangle Angle Sum Theorem):

  • Statement: For every triangle, the sum of the measures of the angles is $180$.

  • Restatement: $m\angle x + m\angle y + m\angle z = 180$

  • Two-Column Proof:

• Corollaries to Theorem 9–13:

  • Corollary 9–13.1: Given a correspondence between two triangles, if two pairs of corresponding angles are congruent, then the third pair of corresponding angles are also congruent.

  • Corollary 9–13.2: The acute angles of a right triangle are complementary ($m\angle A + m\angle C = 90$).

  • Corollary 9–13.3: For any triangle, the measure of an exterior angle is equal to the sum of the measures of its remote interior angles ($m\angle BCD = m\angle A + m\angle B$).

• Angle Measurement Exercise:

• In the diagram with given angles $28^\circ$, $82^\circ$, $47^\circ$, $45^\circ$: Unknown angle measures are solved directly using exterior angle theorems and angle sum relations.

Quadrilaterals

Definitions and Properties of Parallelograms

• Definition of Quadrilateral:

  • Let $A, B, C,$ and $D$ be four points of the same plane. If no three of these points are collinear, and the segments $\overline{AB}, \overline{BC}, \overline{CD},$ and $\overline{DA}$ intersect only at their endpoints, then the union of these four segments is a quadrilateral (denoted $\square ABCD$).

  • Convex Quadrilateral: A quadrilateral where no two vertices lie on opposite sides of a line containing any side.

  • Opposite Sides: Two sides that do not intersect.

  • Consecutive Sides: Two sides sharing a common endpoint.

  • Opposite Angles: Two angles that do not share a side.

  • Consecutive Angles: Two angles sharing a side.

  • Diagonal: A segment joining two non-consecutive vertices.

• Special Quadrilaterals Definitions:

  • Parallelogram: A quadrilateral in which both pairs of opposite sides are parallel.

  • Trapezoid: A quadrilateral in which one and only one pair of opposite sides are parallel. (Parallel sides are bases; non-parallel sides are legs; segment connecting leg midpoints is the median).

  • Isosceles Trapezoid: A trapezoid whose legs are congruent.

• Theorems on Parallelogram Properties:

  • Theorem 9–14: Each diagonal separates a parallelogram into two congruent triangles ($\square ABCD \implies \Delta ABC \cong \Delta CDA$).

  • Theorem 9–15: In a parallelogram, any two opposite sides are congruent ($\overline{AB} \cong \overline{CD}$ and $\overline{AD} \cong \overline{BC}$).

  • Corollary 9–15.1: If two lines are parallel, then all points of each line are equidistant from the other line.

  • Theorem 9–16: In a parallelogram, any two opposite angles are congruent ($\angle A \cong \angle C$ and $\angle B \cong \angle D$).

  • Theorem 9–17: In a parallelogram, any two consecutive angles are supplementary ($m\angle A + m\angle B = 180$).

  • Theorem 9–18: The diagonals of a parallelogram bisect each other.

• Conditions Guaranteeing Parallelograms:

  • Theorem 9–19: Given a quadrilateral in which both pairs of opposite sides are congruent. Then the quadrilateral is a parallelogram.

  • Theorem 9–20: If two sides of a quadrilateral are parallel and congruent, then the quadrilateral is a parallelogram.

  • Theorem 9–21: If the diagonals of a quadrilateral bisect each other, then the quadrilateral is a parallelogram.

Midline Theorem, Special Parallelograms, and Right Triangles

• Theorem 9–22 (The Midline Theorem):

  • Statement: The segment between the midpoints of two sides of a triangle is parallel to the third side and half as long.

  • Restatement: Given $\Delta ABC$, if $D$ and $E$ are midpoints of $\overline{AB}$ and $\overline{BC}$, then $\overline{DE} \parallel \overline{AC}$ and $DE = \frac{1}{2}AC$

  • Two-Column Proof:

• Special Parallelograms Definitions:

  • Rhombus: A parallelogram all of whose sides are congruent.

  • Rectangle: A parallelogram all of whose angles are congruent.

  • Square: A rectangle all of whose sides are congruent.

• Theorems on Special Parallelograms:

  • Theorem 9–23: If a parallelogram has one right angle, then it has four right angles, and the parallelogram is a rectangle.

  • Theorem 9–24: In a rhombus, the diagonals are perpendicular to one another ($\overline{QS} \perp \overline{RT}$).

  • Theorem 9–25: If the diagonals of a quadrilateral bisect each other and are perpendicular, then the quadrilateral is a rhombus.

• Theorems on Right Triangles:

  • Theorem 9–26: The median to the hypotenuse of a right triangle is half as long as the hypotenuse ($AD = \frac{1}{2}BC$).

  • Theorem 9–27 (The 30–60–90 Triangle Theorem): If an acute angle of a right triangle has measure $30^\circ$, then the opposite side is half as long as the hypotenuse ($AB = \frac{1}{2}BC$).

  • Theorem 9–28 (Converse of 30–60–90 Triangle Theorem): If one leg of a right triangle is half as long as the hypotenuse, then the opposite angle has measure $30^\circ$

Polygonal Regions

Area Postulates and Formulas

• Definitions:

  • Triangular Region: The union of a triangle and its interior.

  • Polygonal Region: The union of a finite number of triangular regions in a plane such that if two intersect, their intersection is either a point or a segment.

• Fundamental Area Postulates:

  • Postulate 19 (The Area Postulate): To every polygonal region there corresponds a unique positive real number (denoted $aR$).

  • Postulate 20 (The Congruence Postulate): If two triangles are congruent, then the triangular regions determined by them have the same area ($\Delta ABC \cong \Delta DEF \implies a\Delta ABC = a\Delta DEF$).

  • Postulate 21 (The Area Addition Postulate): If two polygonal regions intersect only in edges and vertices (or do not intersect at all), then the area of their union is the sum of their areas ($aR = aR_1 + aR_2$).

  • Postulate 22 (The Unit Postulate): The area of a square region is the square of the length of its edge ($aR = s^2$).

• Area Formulas for Quadrilaterals and Triangles:

  • Theorem 11–1 (Rectangle): $aR = b \cdot h$

  • Theorem 11–2 (Right Triangle): $aR = \frac{1}{2}(a \cdot b)$

  • Theorem 11–3 (General Triangle): $aR = \frac{1}{2}(b \cdot h)$

  • Theorem 11–4 (Trapezoid): $aR = \frac{1}{2}h(b_1 + b_2)$

  • Theorem 11–5 (Parallelogram): $aR = b_1 \cdot h_1 = b_2 \cdot h_2$

  • Theorem 11–6: If two triangles have the same base $b$ and altitude $h$, then they have the same area.

  • Theorem 11–7: If two triangles have the same altitude $h$, then the ratio of their areas is equal to the ratio of their bases ($\frac{a\Delta_1}{a\Delta_2} = \frac{b_1}{b_2}$).

The Pythagorean Theorem and Special Triangles

• Theorem 11–8 (The Pythagorean Theorem):

  • Statement: In a right triangle, the square of the hypotenuse is equal to the sum of the squares of the legs ($c^2 = a^2 + b^2$).

• Travel Problem Application:

  • Problem: A man travels $1\,km$ north, $2\,km$ east, $3\,km$ north, and $4\,km$ east. Find total distance from starting point.

• Solution:

  • Total North displacement $y = 1 + 3 = 4\,km$.

  • Total East displacement $x = 2 + 4 = 6\,km$.

  • Distance $AB = \sqrt{4^2 + 6^2} = \sqrt{16 + 36} = \sqrt{52} \approx 7.2\,km$.

  • Theorem 11–9 (Converse of the Pythagorean Theorem):

• Statement: If the square of one side of a triangle is equal to the sum of the squares of the other two sides, then the triangle is a right triangle, with its right angle opposite the longest side.

  • Pythagorean Triples: A set of positive integers $a, b, c$ satisfying $c^2 = a^2 + b^2$ (e.g., $\mathbf{\{3, 4, 5\}}$, $\mathbf{\{5, 12, 13\}}$, $\mathbf{\{8, 15, 17\}}$, $\mathbf{\{7, 24, 25\}}$).

  • Special Triangle Theorems:

• Theorem 11–10 (Isosceles Right Triangle Theorem): In an isosceles right triangle ($45^\circ-45^\circ-90^\circ$), the hypotenuse is $\sqrt{2}$ times as long as each leg ($c = a\sqrt{2}$).

• Theorem 11–11 (Converse of Isosceles Right Triangle Theorem): If the base of an isosceles triangle is $\sqrt{2}$ times as long as each of the two congruent sides, then the angle opposite the base is a right angle.

• Theorem 11–12 (30–60–90 Longer Leg Theorem): In a $30^\circ-60^\circ-90^\circ$ triangle, the longer leg (opposite the $60^\circ$ angle) is $\frac{1}{2}\sqrt{3}$ times as long as the hypotenuse ($b = \frac{1}{2}\sqrt{3} \cdot c$).

Similarity

Proportionality and Basic Similarity Theorems

• Definition of Geometric Similarity: Two geometric figures are similar ($\sim$) if they have exactly the same shape, but not necessarily the same size. Squares, circles, and equilateral triangles are ALWAYS similar.

• Proportional Sequences and Geometric Mean:

  • Proportional Sequences: Sequences $a, b, c, \dots$ and $p, q, r, \dots$ are proportional ($a, b, c, \dots \sim p, q, r, \dots$) if $\frac{a}{p} = \frac{b}{q} = \frac{c}{r} = \dots$

  • Theorem 12–1: Proportionality between sequences is an equivalence relation.

  • Geometric Mean Definition: If $a, b, c$ are positive numbers and $\frac{a}{b} = \frac{b}{c}$, then $b$ is the geometric mean between $a$ and $c$ ($b = \sqrt{a \cdot c}$).

  • Arithmetic Mean Definition: $\frac{1}{2}(a + c)$.

• Proportionality and Similarity Theorems:

  • Definition of Similar Triangles: Corresponding angles are congruent, and corresponding sides are proportional.

  • Theorem 12–2 (The Basic Proportionality Theorem): If a line parallel to one side of a triangle intersects the two other sides in distinct points, then it cuts off segments which are proportional to these sides.

  • Theorem 12–3 (Converse of the Basic Proportionality Theorem): If a line intersects two sides of a triangle and cuts off segments proportional to these two sides, then it is parallel to the third side.

  • Theorem 12–4 (The AAA Similarity Theorem): If corresponding angles of two triangles are congruent, the correspondence is a similarity.

  • Corollary 12–4.1 (The AA Corollary): If two pairs of corresponding angles are congruent, the triangles are similar.

  • Corollary 12–4.2: If a line parallel to one side of a triangle intersects the other two sides in distinct points, then it cuts off a triangle similar to the given triangle ($\Delta CDE \sim \Delta CAB$).

Advanced Similarity Theorems and Right Triangle Similarities

• Equivalence and Proportional Similarity Theorems:

  • Theorem 12–5: Similarity between triangles is an equivalence relation.

  • Corollary 12–5.1: If $\Delta ABC \sim \Delta DEF$ and $\Delta DEF \cong \Delta GHI$, then $\Delta ABC \sim \Delta GHI$.

  • Theorem 12–6 (The SAS Similarity Theorem): If two pairs of corresponding sides are proportional, and the included angles are congruent, the triangles are similar.

  • Theorem 12–7 (The SSS Similarity Theorem): If corresponding sides of two triangles are proportional, the triangles are similar.

• Right Triangle Altitude Theorems:

  • Theorem 12–8: In any right triangle, the altitude to the hypotenuse separates the triangle into two triangles which are similar to each other and to the original triangle ($\Delta ABC \sim \Delta ACD \sim \Delta CBD$).

  • Theorem 12–9 (Geometric Mean Theorems in Right Triangles):

  1. The altitude $h$ is the geometric mean of the segments into which it separates the hypotenuse: $h = \sqrt{x \cdot y}$.

  2. Each leg is the geometric mean of the hypotenuse and the segment of the hypotenuse adjacent to the leg: $a = \sqrt{x(x + y)}$ and $b = \sqrt{y(x + y)}$.

• Applied Right Triangle Problem:

  • Scenario: Police station ($P$) is due north of Elsa's house ($E$), Hospital ($H$) is due east. Fire station ($F$) is on the highway at the shortest distance from Elsa's house. $PF = 40\,m$ and $FH = 90\,m$.

• Solutions:

  • a. Distance from Elsa's house to Fire station ($a = h$): $a = \sqrt{40 \cdot 90} = \sqrt{3600} = 60\,m$

  • b. Distance to Police station ($b$): $b = \sqrt{40(40 + 90)} = \sqrt{5200} \approx 72\,m$

  • c. Distance to Hospital ($c$): $c = \sqrt{90(40 + 90)} = \sqrt{11700} \approx 108\,m$

  • Theorem 12–10 (Area Ratio of Similar Triangles):

• Statement: If two triangles are similar, then the ratio of their areas is the square of the ratio of any two corresponding sides:     $\frac{a\Delta ABC}{a\Delta A'B'C'} = \left(\frac{a}{a'}\right)^2 = \left(\frac{b}{b'}\right)^2 = \left(\frac{c}{c'}\right)^2$