The bases of an isosceles trapezoid are 30 and 10. The straight lines connecting the middle of the larger base

| February 23, 2021 | education

The bases of an isosceles trapezoid are 30 and 10. The straight lines connecting the middle of the larger base with the ends of the smaller base intersect the diagonals at points M and N. Find the length of the segment MN.

Let us prove the similarity of the triangles AMK and BCM. Angle AMK = BMC as vertical angles. Angle МАК = ВСМ as criss-crossing angles at the intersection of parallel straight lines AD and BC secant AC. Then the triangles АМК and ВСМ are similar in two angles.

Then: AK / ВС = KM / ВM.

KM / ВM = 15/10 = 3/2.

Similarly, we prove the similarity of the triangles KНD and ВСН. Then:

DK / ВС = KН / CH.

KН / BН = 15/10 = 3/2.

Since the coefficient of similarity is the same in both cases, the following relation is true:

KM / ВM = KН / CH = 3/2.

KM = KН, and CH = СK – KН = ВK – KM, then:

KM / ВM = KM / (ВK – KM) = 3/2.

3 * (ВK – KM) = 2 * KM.

3 * ВK = 2 * KM + 3 * KM.

ВK / KM = 5/3.

Consider triangles ВСК and MНK, which are similar in two angles, the angle K is common for them, and the angle КВС = KMН as the corresponding angles at the intersection of parallel ВC and MН secant ВK.

Then BC / MН = ВK / KM = 5/3.

BC / MH = 5/3.

10 / MH = 5/3.

MH = 10 * 3/5 = 6 cm.

Answer: MH = 6 cm.



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