Answer:
Option b.
Step-by-step explanation:
Given : [tex]\Delta SRQ[/tex] is right angled triangle . An altitude is drawn from point R to point T on side S Q to form a right angle. The length of S T is 9 and the length of T Q is 16. The length of S R is x.
In the figure, [tex]\angle SRQ=90^{\circ}[/tex] and RT is perpendicular to SQ.
We know that in a right angled triangle if a perpendicular is drawn from the vertex of the right angle to the hypotenuse then triangles on both sides of the perpendicular are similar to each other and to the whole triangle .
Therefore , [tex]\Delta STR\sim \Delta SRQ[/tex]
Also, we know that if two triangles are similar then their sides are proportional .
[tex]\frac{ST}{SR}=\frac{SR}{SQ}\\\frac{9}{x}=\frac{x}{25}\\x^2=25\times 9\\x=5\times 3\\=15[/tex]
So, option b. is correct
The value of x which represents the length of the line segment SR is given by: Option B: 15 units
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]
where |AB| = length of line segment AB. (AB and BC are rest of the two sides of that triangle ABC, AC being the hypotenuse).
For this case, referring to the figure corresponding to the given problem statement, we get that:
There are three right angled triangles.
From triangle SRQ, we get:
[tex]x^2 + z^2 = 25^2\\[/tex] -- (1)
From triangle STR, we get:
[tex]x^2 = y^2 + 9^2[/tex] -- (2)
From triangle QTR, we get:
[tex]z^2 = 16^2 + y^2[/tex] --(3)
Putting values from eq. 2 and 3 in first, we get:
[tex]x^2 + z^2 = 25^2\\\\y^2 + 9^2 + 16^2 + y^2 = 25^2\\y^2 = \dfrac{625 - 256 - 81}{2} = 144\\\\y = 12[/tex]
(y cannot be -12(whose sq. is 144 too) as y represents length, a non-negative quantity)
Putting this value in eq. 2, we get:
[tex]x^2 = y^2 + 9^2\\\\x = \sqrt{144 + 81} = \sqrt{225} = 15[/tex]
Thus, the value of x which represents the length of the line segment SR is given by: Option B: 15 units
Learn more about Pythagoras theorem here:
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