Question

1) Proiectiile catetelor unui triunghi dreptunghic pe ipotenuza au lungimile 9 cm si 25 cm. Aflati lungimea inaltimii din varful unghiului drept. 2) Intr-un triunghi dreptunghic lungimea ipotenuzei este de 34 cm , iar lungimile proiectiilor catetelor pe ipotenuza sunt direct proportionale cu numerele 0,(6) si 0,75. Calculati lungimea inaltimii corespunzatoare ipotenuzei

Answer 1

1) 15cm

2) left projection/h = h/right projection

Explanation:

Question:

1) The projections of the legs of a right triangle on the hypotenuse have lengths of 9 cm and 25 cm. Find the length of the height at the top of the right angle.

2) In a right triangle the length of the hypotenuse is 34 cm, and the lengths of the projections of the legs on the hypotenuse are directly proportional to the numbers 0, (6) and 0.75. Calculate the length of the height corresponding to the hypotenuse.

Solution

1) The length of the height of a right angle triangle is also called the altitude.

Since there are no diagrams in the question, I sent a diagram of the right angle as an attachment to the solution.

The projections of the legs are 25cm and 9cm.

Hence, the longer projection length AD = 25cm and the shorter projection length DB = 9cm

In a right triangle, the altitude (height) drawn from the right angle to the hypotenuse divides the hypotenuse into two segments. The length of the altitude is the

geometric mean of these two segments (the two projections) and it's given by:

left projection/h = h/right projection

AD/h = h/DB

25/h = h/9

Cross multiply

h^2 = 25×9

h =√225 = 15cm

The length of the height at the top of the triangle = 15cm

2) Length of hypotenuse = 34

From the question, the lengths of the projections of the legs on the hypotenuse are directly proportional to the numbers 0, (6) and 0.75.

There is an error with the figures because the sum of the length of the projection of the legs should be equal to the hypotenuse but it isn't in this case.

To calculate the length of the height corresponding to the hypotenuse, we would use the same formula above.

left projection/h = h/right projection

To find each leg using question 1 above, each leg of the triangle is the mean proportional between the hypotenuse and the part of the hypotenuse directly below the leg.

Hypotenuse =34cm

Hyp/leg = leg/part

To find leg y, part for leg y = 25cm

34/y = y/25

y^2 = 34×25 = 850

y = √850 = 29.2cm

To find leg x, part for leg x = 9cm

34/y = y/9

y^2 = 34×9 = 306

y = √306 = 17.5cm