Question

1. if a triangle is divided into two smaller triangles by an altitude and all three triangles are similar, what theorem could this be used to prove?

a. sas criterion
b. hl criterion
c. sss criterion
d. pythagorean theorem

1. which criterion requires two angles of one triangle to be congruent to two angles of another triangle for the triangles to be similar?

a. sss
b. aa
c. asa
d. sas

1. in △xyz, the angle bisector of ∠yxz intersects yz at p. if xy = 8, xz = 12, yp = 4, and pz = 6, verify the angle bisector theorem.

a. \\(\frac{yp}{pz}=0.6, \frac{xy}{xz}=1.25\\)
b. \\(\frac{yp}{pz}=0.75, \frac{xy}{xz}=0.75\\)
c. \\(\frac{yp}{pz}=0.67, \frac{xy}{xz}=0.67\\)
d. \\(\frac{yp}{pz}=1, \frac{xy}{xz}=1.5\\)

1. in △abc, d and e are midpoints of ab and ac. if ab = 12 and ac = 16, what is the length of de?

a. 6
b. 9
c. 10
d. 8

1. in triangle abc, if ab = 10 units, bc = 6 units, and ac = 6 units, what type of triangle is triangle abc?

a. isosceles
b. right
c. scalene
d. equilateral

1. in △xyz, ∠xyz = 90°, and yw is the altitude to xz. if xw = 9 and wz = 16, what is the value of xz?

a. 30
b. 20
c. 35
d. 25

1. in △pqr, ∠pqr = 90°, and qs is the altitude to pr. if pq = 6 and qr = 8, what is the length of qs if pr = 10?

Explanation:

Step1: Recall the Angle Bisector Theorem

The Angle Bisector Theorem states that if a bisector of an angle of a triangle divides the opposite side into segments proportional to the adjacent sides. That is, \(\frac{YP}{PZ}=\frac{XY}{XZ}\)

Step2: Calculate \(\frac{YP}{PZ}\) and \(\frac{XY}{XZ}\)

Given \(XY = 8\), \(XZ=12\), \(YP = 4\), and \(PZ = 6\)

• Calculate \(\frac{YP}{PZ}\): \(\frac{YP}{PZ}=\frac{4}{6}=\frac{2}{3}\approx0.67\)
• Calculate \(\frac{XY}{XZ}\): \(\frac{XY}{XZ}=\frac{8}{12}=\frac{2}{3}\approx0.67\)

Answer:

c. \(\frac{YP}{PZ}=0.67,\frac{XY}{XZ}=0.67\)