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Difficulty: MediumPythagorean Theorem and Special Right Triangles

In isosceles trapezoid ABCDABCD, the shorter base ABAB measures 77 units and the longer base CDCD measures 1717 units. The congruent legs ADAD and BCBC each form a 45∘45^\circ angle with base CDCD. What is the length of diagonal ACAC?

  1. 1313Answer
  2. B
    1717
  3. C
    525\sqrt{2}
  4. D
    119\sqrt{119}
  5. E
    169169

Answer

13 units
Dropping altitude APAP perpendicular to base CDCD divides base CDCD into DP=5DP = 5 units and PC=12PC = 12 units. Since triangle APDAPD is a 45∘−45∘−90∘45^\circ-45^\circ-90^\circ right triangle, height AP=DP=5AP = DP = 5. Right triangle APCAPC has legs AP=5AP = 5 and PC=12PC = 12, making hypotenuse AC=52+122=13AC = \sqrt{5^2 + 12^2} = 13.

Step-by-Step Solution

1
Find the length of the base projection segment for the isosceles trapezoid.
Segment DP=5DP = 5 units.
Draw altitude APAP perpendicular to CDCD. Because trapezoid ABCDABCD is isosceles, the projection DP=CD−AB2=17−72=5DP = \frac{CD - AB}{2} = \frac{17 - 7}{2} = 5.
2
Determine the altitude of the trapezoid using special right triangle properties.
Altitude AP=5AP = 5 units.
Triangle APDAPD is a 45∘−45∘−90∘45^\circ-45^\circ-90^\circ right triangle, so its legs are congruent (AP=DP=5AP = DP = 5).
3
Calculate the length of the remaining base segment in right triangle APCAPC.
Segment PC=12PC = 12 units.
Segment PC=CD−DP=17−5=12PC = CD - DP = 17 - 5 = 12.
4
Apply the Pythagorean Theorem to right triangle APCAPC to solve for diagonal ACAC.
Diagonal AC=13AC = 13 units.
AC=AP2+PC2=52+122=25+144=169=13AC = \sqrt{AP^2 + PC^2} = \sqrt{5^2 + 12^2} = \sqrt{25 + 144} = \sqrt{169} = 13.

Key Concept

Pythagorean Theorem and Special Right Triangles
Estimated Time:1m 15s
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Topics

Pythagorean Theorem and Special Right Triangles

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