Two questions here:
1. In a right-angled triangle ABC, catheter AC is 9.4 cm long and catheter BC is 6.2 cm long. D is a point on the catheter AC so that DA = DB. Determine the length of the CD.
2. In a right-angled triangle, the ratio of catheters is 2: 3. The height towards the hypotenuse is 2.0 cm long. Determine the length of the hypotenuse.
plural of cathetus is catheti, not catheter !
Haven't heard "catheter" used as being a side in a right-angled triangle in a long time ....
I made the diagram, let AD = x, then DC = 9.4-x
(9.4-x)^2 + 6.2^2 = x^2
88.36 - 18.8x + x^2 + 38.44 = x^2
18.8x = 126.8
x = appr 6.74
so DC = ...
b) Let the 2 legs be 2x and 3x
if h is the hypotenuse:
h^2 = 4x^2 + 9x^2
h = √13 x
Look at your sketch, you have similar triangles, so set up ratios of corresponding sides
I see: 2/(2x) = 3x/√13 x = 3/√13
6x = 2√13
x = √13/3
then the hypotenuse = √13(√13/3) = 13/3
catheter?
AD^2 = 6.2^2 + 9.4^2 = 38.44 + 88.36 = 126.8
so
AD = 11.26
let CD = q
let DA = DB =p
q + p = 9.4 so p = 9.4-q
q^2 + 6.2^2 = p^2 = (9.4-q)^2
q^2 + 38.44 = 88.36 - 18.8 q + q^2
18.8 q = 49.92
q = 2.65
2. In a right-angled triangle, the ratio of catheters is 2: 3. The height towards the hypotenuse is 2.0 cm long. Determine the length of the hypotenuse.
====================================================
Get that Catheter out of me !
2 a and 3 a legs so
hypotenuse = a * sqrt 13
sin of one angle in the corner = 2/sqrt 13 = 2/3a
3 a = sqrt 13
a =sqrt 13/3
hyp = a sqrt 13 = 13/3
To find the length of CD in question 1, we can use the Pythagorean theorem. The Pythagorean theorem states that in a right-angled triangle, the square of the length of the hypotenuse is equal to the sum of the squares of the lengths of the other two sides.
In this case, we know the lengths of the catheters AC and BC. We want to find the length of CD, which is a segment on the catheter AC.
Since DA = DB, we can divide the catheter AC into two equal segments: AD and DC. Let's assume that AD = x. That means DC = x as well.
We can now set up the equation using the Pythagorean theorem:
AC^2 = AD^2 + DC^2
Substituting the given values:
(9.4 cm)^2 = (x)^2 + (x)^2
Simplifying:
88.36 cm^2 = 2x^2
Divide by 2:
44.18 cm^2 = x^2
Taking the square root of both sides:
x ≈ 6.64 cm
Since CD = DC = x, the length of CD is approximately 6.64 cm.
For question 2:
To find the length of the hypotenuse, we can again use the Pythagorean theorem. The Pythagorean theorem states that the square of the length of the hypotenuse is equal to the sum of the squares of the lengths of the other two sides.
Let's assume that the shorter catheter has a length of 2x and the longer catheter has a length of 3x. The height towards the hypotenuse is given as 2.0 cm.
Using the Pythagorean theorem:
(2x)^2 + (3x)^2 = hypotenuse^2
4x^2 + 9x^2 = hypotenuse^2
13x^2 = hypotenuse^2
Taking the square root of both sides:
sqrt(13x^2) = hypotenuse
Simplifying:
sqrt(13) * x = hypotenuse
Since the length of the height towards the hypotenuse is given as 2.0 cm, we can equate it to x to find x:
2.0 cm = x
Now, we can calculate the length of the hypotenuse:
hypotenuse = sqrt(13) * 2.0 cm
Calculating:
hypotenuse ≈ 5.7 cm
Therefore, the length of the hypotenuse is approximately 5.7 cm.