Question

Prove the Converse of the Pythagorean Theorem using similar triangles. The Converse of the Pythagorean Theorem states that when the sum of the squares of the lengths of the legs of the triangle equals the squared length of the hypotenuse, the triangle is a right triangle. Be sure to create and name the appropriate geometric figures. (10 points)

Let ACB be a triangle where segment AB is labeled c, segment BC is labeled a, and segment CA is labeled b. Using a compass and straightedge, construct right triangle XZY with legs XZ and ZY congruent to legs AC and CB , respectively. Let XZ be labeled y, ZY labeled x, and YX labeled z. Therefore, y = b and x = a. Because triangle XZY was constructed as a right triangle, x^2 + y^2 = z^2 by the _____ _____. By _____, a^2 + b^2 = z^2. According to the given information, a^2 + b^2 = c^2. By the _____ _____ _____ _____, c^2 = z^2. Use the _____ _____ _____ _____ _____ to take the square root of both sides of the equation to result in the equation c = z. By the ______ _____, triangles ACB and XZY are congruent. Consequently, by _____, triangle ACB is a right triangle.


2. Look at the figure below:




Copy the Statements and Reasons into your answer box.

Statements
Reasons
Angle PQS in triangle PQS is congruent to angle PRQ in triangle PRQ


Angle QPS is common to both triangle PRQ and triangle PQS


Triangle PRQ is similar to triangle PQS




Look at the figure.





3. Annie walks 15 feet away from her house and places a mirror on the ground. She backs 4 feet away from the mirror so that she can see the tip of the roof. Annie’s eyes are 5 feet above the ground. Annie and the house are both perpendicular to the ground. The angles between the top of the house, the mirror, and the ground and between Annie’s eyes, the mirror, and the ground are congruent as shown in the image:




What is the height of the house? Show your work and explain your reasoning in complete sentences.

The triangle formed by the person, the mirror, and the ground is similar to the triangle formed by the house, the mirror, and the ground. The distance between the mirror and the person is 4 feet. The distance between the mirror and the house is 15 feet. The distance between the ground and the person’s eye level is 5 feet. Let x = the height of the house.

Answer 1

EF = BC = a ÐF is a right angle. FD = CA = b triangle EF = BC = a angle F is a right angle. FD = CA = b In triangle DEF, By Pythagoras Theorem, a2 + b2 = c2 the given AB=c= a^2 + b^2 square root Theorefore AB = DE But by construction, BC = EF and CA = FD triangle ABC congruent to DEF (S.S.S

We have given that Δ ABC is similar to Δ CBD and Δ ABC is similar to Δ ACD according to the attached picture.

Because of the similarity the corresponding sides are proportional. Then a/c = f/a

and b/e = c/b. If we cross multiply, we'll get

a² = cf and b² = ce

If we add these equalities together, we'll get

a² + b² = c (f + e)

From the beginning, we know that c = e+f

Then, the final result is:

a² + b² = c²