In a right-angled triangle ABC: CD is the height drawn by the hypotenuse. One of the legs is 8, and its projection onto

| March 25, 2021 | education

In a right-angled triangle ABC: CD is the height drawn by the hypotenuse. One of the legs is 8, and its projection onto the hypotenuse is 6.4. Find 5 * (r1 + r2), where r1 and r2 are the radii of the circles inscribed in the ACD and BCD triangles

In a right-angled triangle BCD, according to the Pythagorean theorem, we determine the length of the leg CD.

CD ^ 2 = BC ^ 2 – BD ^ 2 = 64 – 40.96 = 23.04.

CD = 4.8 cm.

Then R1 = (BD + CD – BC) / 2 = (6.4 + 4.8 – 8) / 2 = 1.6 cm.

The height of the CD is drawn from the top of the right angle, then CD ^ 2 = AD * BD.

AD = CD ^ 2 / BD = 23.04 / 6.4 = 3.6 cm.

By the Pythagorean theorem, from a right-angled triangle ACD, AC ^ 2 = CD ^ 2 + AD ^ 2 = 23.04 + 12.96 = 36.

AC = 6 cm.

Then R2 = (AD + CD – AC) / 2 = (3.6 + 4.8 – 6) / 2 = 1.2 cm.

5 * (R1 + R2) = 5 * (1.6 + 1.2) = 5 * 2.8 = 14.

Answer: 5 * (R1 + R2) = 14.



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