Math

QuestionIn line BDCB D C, given AB=40A B=40, BC=32B C=32, AC=24A C=24, and CD=7C D=7, determine if C\angle C is right and find ADA D.

Studdy Solution

STEP 1

Assumptions1. In the figure, BDCB D C is a straight line. . The lengths of the sides are AB=40,BC=32,AC=24A B=40, B C=32, A C=24 and CD=7C D=7.

STEP 2

To determine if C\angle C is a right angle, we can use the Pythagorean theorem. This theorem states that in a right-angled triangle, the square of the length of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the lengths of the other two sides.
AB2=AC2+BC2A B^2 = A C^2 + B C^2

STEP 3

Now, plug in the given values for AB,ACA B, A C and BCB C to check if the equation holds true.
402=242+32240^2 =24^2 +32^2

STEP 4

Calculate the squares of the lengths.
1600=576+10241600 =576 +1024

STEP 5

Add the squares of ACA C and BCB C.
1600=16001600 =1600Since both sides of the equation are equal, C\angle C is a right angle.

STEP 6

To find ADA D, we need to use the Pythagorean theorem again. This time, we consider triangle ACDA C D.
AC2=AD2+CD2A C^2 = A D^2 + C D^2

STEP 7

Rearrange the equation to solve for ADA D.
AD2=AC2CD2A D^2 = A C^2 - C D^2

STEP 8

Plug in the given values for ACA C and CDC D.
AD2=24272A D^2 =24^2 -7^2

STEP 9

Calculate the squares of the lengths.
AD2=57649A D^2 =576 -49

STEP 10

Subtract to find AD2A D^2.
AD2=527A D^2 =527

STEP 11

Take the square root of both sides to solve for ADA D.
AD=527A D = \sqrt{527}

STEP 12

Calculate the square root.
AD22.94A D \approx22.94So, AD22.94A D \approx22.94 units.

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