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Difficulty: MediumTriangles: Properties, Perimeter, and Area

In triangle ABCABC, the length of side ABAB is 1616 units and the length of side ACAC is 1313 units. If the area of triangle ABCABC is 9696 square units and the altitude from vertex CC to side ABAB intersects the line segment ABAB at point DD, what is the length of segment ADAD?

  1. 55Answer
  2. B
    1111
  3. C
    1212
  4. D
    77
  5. E
    11

Answer

The length of segment ADAD is 55 units.
The correct answer is 55. First, find altitude CDCD using the triangle area equation: Area=12×base×height  ⟹  96=12×16×CD  ⟹  CD=12\text{Area} = \frac{1}{2} \times \text{base} \times \text{height} \implies 96 = \frac{1}{2} \times 16 \times CD \implies CD = 12. Next, because CD⊥ABCD \perp AB, triangle ADCADC is a right triangle with hypotenuse AC=13AC = 13 and leg CD=12CD = 12. Applying the Pythagorean theorem yields AD=132−122=25=5AD = \sqrt{13^2 - 12^2} = \sqrt{25} = 5.

Step-by-Step Solution

1
Calculate the length of altitude CDCD using the triangle area formula.
CD=12CD = 12 units.
The area of triangle ABCABC is given by Area=12×AB×CD\text{Area} = \frac{1}{2} \times AB \times CD. Substituting the given values: 96=12×16×CD  ⟹  96=8×CD  ⟹  CD=1296 = \frac{1}{2} \times 16 \times CD \implies 96 = 8 \times CD \implies CD = 12.
2
Apply the Pythagorean theorem in right triangle ADCADC to find ADAD.
AD=5AD = 5 units.
Altitude CDCD is perpendicular to ABAB, forming right triangle ADCADC with hypotenuse AC=13AC = 13 and leg CD=12CD = 12. By the Pythagorean theorem, AD2+CD2=AC2  ⟹  AD2+122=132  ⟹  AD2+144=169  ⟹  AD2=25  ⟹  AD=5AD^2 + CD^2 = AC^2 \implies AD^2 + 12^2 = 13^2 \implies AD^2 + 144 = 169 \implies AD^2 = 25 \implies AD = 5.

Key Concept

Triangles: Properties, Perimeter, and Area
Estimated Time:1m 30s
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Topics

Triangles: Properties, Perimeter, and Area

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