The bases of an isosceles trapezoid are 16 and 96, the lateral side is 58. Find the length of the diagonal of the trapezoid.

| July 21, 2021 | education

Given: trapezoid, where AD || BC, BC = 19, AD = 96, AB = CD = 58.
It is required to determine: the length of the trapezoid diagonal.
Let us use the following property of an isosceles trapezoid. The angles at any base are ∠BAD = ∠CDA, ∠ABC = ∠DCB.
Let’s draw a diagonal AC and apply the cosine theorem to the resulting two triangles ABC and CDA: AC² = AB² + BC² – 2 * AB * BC * cos∠ABC = 3620 – 1856 * cos∠ABC and AC² = AD² + CD² – 2 * AD * CD * cos∠CDA = 12580 – 11136 * cos∠CDA.
We have: cos∠ABC = (3620 – AC²) / 1856 and cos∠CDA = (12580 – AC²) / 11136.
Since AD ​​|| BC, and the line AC intersects them, then the sum of the one-sided angles ∠ABC and ∠BAD is 180 °, that is, ∠ABC + ∠BAD = 180 ° or ∠ABC + ∠CDA = 180 °, whence ∠CDA = 180 ° – ∠ ABC.
Recall the following reduction formula cos (180 ° – α) = –cosα and note that cos∠CDA = cos (180 ° – ∠ABC) = –cos∠ABC.
Therefore, (12580 – AC²) / 11136 = – (3620 – AC²) / 1856 or 12580 – AC² = 6 * (AC² –3620), whence AC² = (12580 + 6 * 3620) / (6 + 1) = 34300 / 7 = 4900, i.e. AC = 70.
Answer: The length of the diagonal of the trapezoid is 70.



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