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Difficulty: MediumTriangles: Properties, Perimeter, and Area

In right triangle ABCABC, the measure of angle CC is 90∘90^\circ. Point DD lies on segment ACAC such that AD=11AD = 11 and BD=13BD = 13. If DC=5DC = 5, what is the perimeter of triangle ABDABD?

Answer: 44

Answer

The perimeter of triangle ABDABD is 4444.
To find the perimeter of triangle ABDABD, we need the lengths of its three sides: ADAD, BDBD, and ABAB. We are given AD=11AD = 11 and BD=13BD = 13. To find ABAB, we first analyze right triangle BCDBCD with right angle at CC, hypotenuse BD=13BD = 13, and leg DC=5DC = 5. Using the Pythagorean theorem, leg BC=132−52=12BC = \sqrt{13^2 - 5^2} = 12. Next, the full leg AC=AD+DC=11+5=16AC = AD + DC = 11 + 5 = 16. In right triangle ABCABC, legs are AC=16AC = 16 and BC=12BC = 12, making hypotenuse AB=162+122=20AB = \sqrt{16^2 + 12^2} = 20. Finally, the perimeter of triangle ABDABD is 11+13+20=4411 + 13 + 20 = 44.

Step-by-Step Solution

1
Find the length of side BCBC using right triangle BCDBCD
BC=12BC = 12
In right triangle BCDBCD, angle C=90∘C = 90^\circ, hypotenuse BD=13BD = 13, and leg DC=5DC = 5. By the Pythagorean theorem, BC=132−52=144=12BC = \sqrt{13^2 - 5^2} = \sqrt{144} = 12.
2
Determine the total length of side ACAC
AC=16AC = 16
Since point DD lies on segment ACAC, AC=AD+DC=11+5=16AC = AD + DC = 11 + 5 = 16.
3
Calculate the hypotenuse ABAB of the main right triangle ABCABC
AB=20AB = 20
In right triangle ABCABC, legs are AC=16AC = 16 and BC=12BC = 12. By the Pythagorean theorem, AB=162+122=256+144=400=20AB = \sqrt{16^2 + 12^2} = \sqrt{256 + 144} = \sqrt{400} = 20.
4
Calculate the perimeter of triangle ABDABD
Perimeter =44= 44
The sides of triangle ABDABD are AD=11AD = 11, BD=13BD = 13, and AB=20AB = 20. Adding these gives 11+13+20=4411 + 13 + 20 = 44.

Key Concept

Pythagorean Theorem and Multi-Step Triangle Properties
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Topics

Triangles: Properties, Perimeter, and Area

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