Answer:
HC = 5.25 units
Step-by-step explanation:
It is given in the ΔABC, ∠ABC = 90°, side BH is the altitude of the triangle.
AB = 9 and AC = 12 units and we have to find the length of HC.
In the right angle triangle ΔABC
[tex]Cosc=\frac{base}{hypotenuse}=\frac{AB}{AC}=\frac{9}{12}[/tex]
Now from ΔABH
[tex]cos c=\frac{AH}{AB}[/tex]
Since cosine of angle is common in both the triangles.
Therefore [tex]\frac{AH}{AB}=\frac{9}{12}[/tex]
[tex]\frac{AH}{9}=\frac{9}{12}[/tex]
[tex]AH=\frac{81}{12}=6.75 units[/tex]
Since HC = AC - AH = 12 - 6.75 = 5.25 units
Therefore answer will be HC = 5.25 units.
You can use Pythagoras theorem and get two equations for this problem. Then use any of the methods to solve that system of linear equation to get the missing values.
The length of HC line segment is 5.25 units approx.
If ABC is a triangle with AC as the hypotenuse and angle B with 90 degrees then we have:
[tex]|AC|^2 = |AB|^2 + |BC|^2[/tex]
(For this case)
Refer to the diagram attached below.
For triangle ABC, we have, by Pythagoras theorem,
[tex]|AC|^2 = |AB|^2 + |BC|^2\\|BC| = \sqrt{12^2 - 9^2} = 8 \text{\:\:(Positive root since length of BC is length thus a non negative quantity)}[/tex]
Let length of AH be x units and length of BH be y units, then length of HC will be 12 - x units.
[tex]|BC|^2 = |HC|^2 + |BH|^2\\64 = (12-x)^2 + y^2\\64 = 144 + x^2 - 24x + y^2\\x^2 + 81 - 24x + y^2 = 0[/tex]
[tex]|AH|^2 + |BH|^2 = 9^2\\x^2 + y^2 = 81[/tex]
Putting this value in the equation we got for triangle BHC, we get
[tex]x^2 + 81 - 24x + y^2 = 0\\(x^2 + y^2) + 81 - 24x = 0\\81 + 81 - 24x = 0\\\\x = \dfrac{162}{24} = 6.75 \; \rm units[/tex]
Putting this value in the equation we got for triangle AHB, we get:
[tex]x^2 + y^2 = 81\\y = \sqrt{81 - x^2}\\\\y = \sqrt{81 - (6.75)^2} \approx 6.28[/tex]
Thus, |BH| = y = 6.28 units approx
and |AH| = x = 6.75 units approx
and |HC| = 12 - x = 5.25 units approx.
Thus,
The length of HC line segment is 5.25 units approx.
Learn more about right angled triangles here:
https://brainly.com/question/4456796
