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The Angle-Bisector Ratio Theorem: Practice

12 multiple-choice questions, progressively harder.

Level 3 · Challenge 0 / 12 answered

Question 1 of 12

1

AD is an internal bisector and $BD/BC=7/18.$ What is AB:AC?

Answer choices for question 1 A $7:18$ B $11:7$ C $7:11$ D $18:7$
2

The true internal bisector meets BC at D0. A given interior D has the same division ratio. Why does this prove $D=D0$ ?

Answer choices for question 2 A Every two interior points are equal B A fixed segment has only one interior point with a given positive division ratio C Equal ratios make BC equal to AB D The two small triangles must first be congruent
3
$\begin{aligned}AB&=12\\AC&=17\end{aligned}$
in a triangle with internal bisector AD. Why is the proposed ordered similarity ABD to ACD impossible?

Answer choices for question 3 A The two equal half-angles are impossible B No two triangles can share a side C An internal bisector must be a midpoint D AB-to-AC would scale by 17/12, but shared AD-to-AD scales by 1
4

A record lists
$\begin{aligned}AB&=6\\AC&=14\\BC&=7\end{aligned}$
and positive side parts in ratio 3:7. Can these measurements describe a nondegenerate triangle with an internal bisector?

Answer choices for question 4 A No; the triangle inequality fails B Yes; positive side parts always guarantee a triangle C Yes; every positive ratio proves all geometry conditions D No; angle bisectors require equal adjacent sides
5

AD internally bisects angle BAC, with
$\begin{aligned}AB&=16\\AC&=26\\BD&=x+1\\DC&=2x-1.\end{aligned}$
Find x.

Answer choices for question 5 A $3$ B $7$ C $5$ D $9$
6

For an internal bisector,
$\begin{aligned}BD&=5\\DC&=8.\end{aligned}$
If AC exceeds AB by 9, find AB.

Answer choices for question 6 A $24$ B $9$ C $15$ D $39$
7

BE internally bisects angle ABC and E is strictly between A,C. Which ordered ratio is correct?

Answer choices for question 7 A $\frac{AE}{EC}=\frac{AB}{BC}$ B $\frac{AE}{EC}=\frac{BC}{AB}$ C $\frac{AE}{AC}=\frac{AB}{BC}$ D $\frac{AE}{EC}=\frac{BE}{BC}$
8

A proposed use of the converse takes D beyond C on line BC. What must be checked before applying this lesson’s converse?

Answer choices for question 8 A Every angle must be acute B The triangle must have a right angle C AD must already be known as a bisector D D must lie strictly between B and C
9

In a nondegenerate triangle, an internal bisector has
$\begin{aligned}AB&=p\\AC&=q\\BC&=L.\end{aligned}$
Which formula gives BD?

Answer choices for question 9 A $\frac{pL}{q}$ B $\frac{qL}{p+q}$ C $\frac{(p+q)L}{p}$ D $\frac{pL}{p+q}$
10

An internal bisector AD meets BC at its midpoint. Which reasoning proves $AB=AC$ ?

Answer choices for question 10 A BD/DC=1, so the ratio theorem gives AB/AC=1 B Every midpoint makes all three sides equal C The two small triangles are automatically similar D A perpendicular-distance theorem is required
11

In the proof, B,A,E are collinear in order, D is interior to BC, and AD is parallel to EC. Which ordered AA similarity follows from the parallel cut?

Answer choices for question 11 A $\triangle BAD\sim\triangle BCE$ B $\triangle BAD\sim\triangle BEC$ C $\triangle BAD\sim\triangle EBC$ D $\triangle BAD\sim\triangle CEB$
12

An explorer reports BD/DC≈0.734 and AB/AC≈0.734. What does that observation establish?

Answer choices for question 12 A Rounded equality alone proves an exact theorem for every triangle B The two small triangles must be congruent C It is consistent with the theorem; the exact claim still needs the geometric proof D The shown ratio is a dilation scale factor

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