Question

In the right ∆ABC, BL is an angle bisector. If LB = 1.2 in and LC = 0.6 in.

Find the distance from L to AB
AND find the measurement of

Answer 1

`\overline {LD}` = 0.6 inch

m∠ABC = 60°

Explanation:

The given parameters are;

The given triangle ΔABC = A right triangle

The angle bisector of ∠CBA = BL

The length of BL = LB = 1.2 in.

The length of LC = 0.6 in.

We have;

`\overline {LB}` ≅
`\overline {LB}` reflexive property

∠ABC = ∠CBL + ∠DBL and ∠CBL ≅ ∠DBL definition of bisected angle

∠CBL = ∠DBL by definition of congruency

∠CLB + ∠CBL = ∠CLB + ∠DBL = 90°, opposite angles of a the right angle triangle are supplementary

∴ ∠CLB = ∠CLB by addition property of equality

ΔLDB ≅ ΔLBC are congruent by the Angle-Side-Angle rule of congruency

∴
`\overline {LD}` ≅
`\overline {LC}` Congruent Parts of Congruent Triangles are Congruent (CPCTC)

`\overline {LD}` =
`\overline {LC}` = 0.6 in. by definition of congruency

`\overline {LD}` = 0.6 in.

sin(m∠CBL) = Opposite side/(Hypotenuse side) =
`\overline {LC}`/
`\overline {LB}` = 0.6/1.6 = 1/2

m∠CBL = sin⁻¹(1/2) = 30°

m∠CBL = m∠DBL = 30°

m∠ABC = m∠CBL + m∠DBL = 30° + 30° = 60°

m∠ABC = 60°