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ALTITUDE ON HYPOTENUSE THEOREM

Learn about applying the altitude on hypotenuse theorem in triangles and solve examples to find missing sides and heights. Detailed explanations provided.

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ALTITUDE ON HYPOTENUSE THEOREM

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  1. ALTITUDE ON HYPOTENUSE THEOREM

  2. Take a triangle PQR, R Q P If an altitude is drawn to the hypotenuse of the right triangle R p q h Q P x y S r Then , Δ PQR ~Δ PSR ~Δ RSQ h² = x X y p² = y X r q² = x X r Where r = x + y

  3. Example 1 : From the figure Find x , h and Y A q=3 cm p=6 cm h x y C B According to altitude on Hypotenuse Theorem p² = y X r → equation 1 q² = xX r → equation 2 h² = xX y → equation 3 r=9 cm In the given figure , p = 3 , r = 9 , q = 6 Substitute the values in the given equation p² = y X r 3² = y X 9 9 = y X 9 y = 9 ÷ 9 y = 1 cm q² = x X r 6² = x X 9 36 = x X 9 x = 36 ÷ 9 x = 4 cm h² = xX y h² = 4 X 1 h² = 4 h = √4 h = 2 cm

  4. Example 2 : If RQ = 15 cm , RP = 20 cm. Find PQ, PS and SQ. R 15 cm 20 cm P Q S Step 1 : To find PQ we apply Pythagoras theorem to Δ PQR, PQ² = RP² + RQ² PQ² = 20² + 15² PQ² = 400 + 225 PQ² = 625 PQ = √625 PQ = 25 cm

  5. Continuation R Step 2 : To find the PS and SQ so we apply altitude on hypotenuse theorem, According to the altitude on Hypotenuse theorem, RP² = PS X PQ → equation 1 RQ² = SQ X PQ → equation 2 Substitute the values from the figure in equation 1 and equation 2, 15 m 20 m P Q S RP² = PS X PQ 20² = PS X 25 400 = PS X 25 400 ÷ 25 = PS PS = 16 RQ² = SQ X PQ 15² = SQ X 25 225 = PS X 25 225 ÷ 25 = PS SQ = 9 AnswerPS = 16cm , SQ = 9cm

  6. Try these 1) In the given figure , find MA and AT. H 8 cm 6 cm M T A 24 cm 2) If XY = 12 m , XZ=16m , YZ = 20 m in the given figure. Find YO and OZ. X 16 m 12 m Y Z O 20 cm

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