Question

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

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

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 { y p } { y z } = 0.6, \frac { x y } { x z } = 1.25 \\)
b. \\( \frac { y p } { y z } = 0.75, \frac { x y } { x z } = 0.75 \\)
c. \\( \frac { y p } { y z } = 0.67, \frac { x y } { x z } = 0.67 \\)
d. \\( \frac { y p } { y z } = 1, \frac { x y } { x z } = 1.5 \\)

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

in triangle abc, if ab = 10 units, bc = 8 units, and ac = 6 units, what type of triangle is triangle abc?
a. isosceles
b. right
c. scalene
d. equilateral

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. 15
d. 25

Explanation:

Step1: Recall the Angle Bisector Theorem

The Angle Bisector Theorem states that \(\frac{YP}{PZ}=\frac{XY}{XZ}\).

Step2: Calculate \(\frac{YP}{PZ}\)

Given \(YP = 4\) and \(PZ=6\), then \(\frac{YP}{PZ}=\frac{4}{6}=\frac{2}{3}\approx0.67\).

Step3: Calculate \(\frac{XY}{XZ}\)

Given \(XY = 8\) and \(XZ = 12\), then \(\frac{XY}{XZ}=\frac{8}{12}=\frac{2}{3}\approx0.67\).

Answer:

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