Answer :
The measure of side BC is 12 units in a right ∆ABC if the altitude CH to the hypotenuse AB intersects angle bisector AL in point D.
What is meant by right angle?
A right angle is defined as an angle that is exactly 90 degrees in geometry and trigonometry. If a ray is positioned so that its terminal is on a line and the angles are also equal, the angles are said to be at right angles. The phrase derives from the Latin angulus rectus, where rectus denotes the upright vertical line that is perpendicular to a horizontal base line.
Perpendicular lines—defined as lines that meet at right angles—and orthogonality—defined as the ability to intersect at right angles—are closely connected and crucial geometrical notions.
Given,
m∠CAL = m∠BAL
m∠C = 90°
AD = 8 cm
DH = 4 cm
We have to apply sine rule for ΔADH,
sinθ=opposite side/hypotenuse
sin(∠DAH)=DH/AD
=4/8
=1/2
∠DAH=sin⁻¹(1/2)
∠DAH=30°
Therefore, ∠CAB=2(∠DAH)
=2(30°)
=60°
Now, we have to apply tangent rule in ∆AHD
tan( ∠DAH)=DH/AH
tan(30°)=4/AH
1/√3=4/AH
AH=4√3
Now, we have to apply cosine rule in ∆AHC
cos(60°)=AH/AC
1/2=4√3/AC
AC=8√3
Now, we have to apply tangent rule in ∆ACB
tan(∠CAB)=opposite side/adjacent side
tan(60°)=BC/AC
√3=BC/8√3
BC=8√3(√3)
BC=24units.
Therefore, the measure of side BC is 12 units in a right ∆ABC if the altitude CH to the hypotenuse AB intersects angle bisector AL in point D.
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