Week 9: Similarity in Right Triangles Answers

Triangle FGH is an isosceles right triangle with a hypotenuse that measures 16 units. An altitude, , is drawn from the right angle to the hypotenuse.

Consider the diagram and the paragraph proof below.Given: Right △ABC as shown where CD is an altitude of the triangleProve: a2 + b2 = c2 Because △ABC and △CBD both have a right angle, and the same angle B is in both triangles, the triangles must be similar by AA. Likewise, △ABC and △ACD both have a right angle, and the same angle A is in both triangles, so they also must be similar by AA. The proportions and are true because they are ratios of corresponding parts of similar triangles. The two proportions can be rewritten as a2 = cf and b2 = ce. Adding b2 to both sides of first equation, a2 = cf, results in the equation a2 + b2 = cf + b2. Because b2 and ce are equal, ce can be substituted into the right side of the equation for b2, resulting in the equation a2 + b2 = cf + ce. Applying the converse of the distributive property results in the equation a2 + b2 = c(f + e).

Triangle S R Q is shown. Angle S R Q is a right angle. An altitude is drawn from point R to point T on side S Q to form a right angle. The length of S T is 9 and the length of T Q is 16. The length of S R is x.

Triangle A B C is shown. Angle A B C is a right angle. An altitude is drawn from point B to point D on side A C to form a right angle. The length of A D is 5 and the length of B D is 12.

ΔQRS is a right triangle.

Triangle X Y Z is shown. Angle Z W X is a right angle. An altitude is drawn from point W to point Y on side Z X to form a right angle. The length of W Y is 4, the length of Y X is a, the length of Z Y is 3, the length of Z W is c, and the length of W X is b.

One leg of an isosceles right triangle measures 5 inches. Rounded to the nearest tenth, what is the approximate length of the hypotenuse?

Triangle MRN is created when an equilateral triangle is folded in half.

Triangle A B C is shown. Angle A C B is a right angle. An altitude is drawn from point C to a point on side A B to form a right angle. The length of A B is 17 and the length of A C is 8.

Triangle Z W X is shown. Angle Z W X is a right angle. An altitude is drawn from point W to point Y on side Z X to form a right angle. The length of W X is b, the length of X Y is a, the length of Y Z is 5, and the length of W Y is 6.