In triangle abc, ac=10, bc=8, angle b =90 degrees and angle bda = 90. how long is line bd
1 answer:
The complete question in the attached figure
we know that
<span>applying the Pythagorean theorem in triangle ABC
</span>AC²=AB²+BC²----------> AB²=AC²-BC²---------> 10²-8²-------> AB²=36-----> AB=6
in the triangle ABC
tan a=BC/AB--------> 8/6--------> tan a=4/3
tan c=AB/BC--------> 6/8--------> tan c=3/4
in the triangle ABD
tan a=BD/AD---------> tan a=BD/x-------> BD=x*tan a
BD=x*(4/3)-----------> equation 1
in the triangle BDC
tan c=BD/DC------------------> BD=(10-x)*tan c------> BD=(10-x)*(3/4)
BD=(10-x)*(3/4)-------------> equation 2
equal 1 and 2
x*(4/3)=(10-x)*(3/4)------> 16x=9*(10-x)--------> 16x=90-9x----> 25x=90
x=90/25-----------> x=3.6
BD=x*(4/3)--------> BD=3.6*(4/3)----------> BD=4.8 units
the answer is BD=4.8 units
we know that
<span>applying the Pythagorean theorem in triangle ABC
</span>AC²=AB²+BC²----------> AB²=AC²-BC²---------> 10²-8²-------> AB²=36-----> AB=6
in the triangle ABC
tan a=BC/AB--------> 8/6--------> tan a=4/3
tan c=AB/BC--------> 6/8--------> tan c=3/4
in the triangle ABD
tan a=BD/AD---------> tan a=BD/x-------> BD=x*tan a
BD=x*(4/3)-----------> equation 1
in the triangle BDC
tan c=BD/DC------------------> BD=(10-x)*tan c------> BD=(10-x)*(3/4)
BD=(10-x)*(3/4)-------------> equation 2
equal 1 and 2
x*(4/3)=(10-x)*(3/4)------> 16x=9*(10-x)--------> 16x=90-9x----> 25x=90
x=90/25-----------> x=3.6
BD=x*(4/3)--------> BD=3.6*(4/3)----------> BD=4.8 units
the answer is BD=4.8 units
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Answer:
Step-by-step explanation:
We can prove that a tangent will always be perpendicular to the radius touching it. So, the other angle in the diagram is .
Because all the angles of a triangle sum to , we have that
.
We combine like terms on the left side to get .
We subtract on both sides to get
.
So, and we're done!
Answer:
- x = 8√3
- y = z = 12√2
Step-by-step explanation:
We presume you want the values of x, y, and z.
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There are two "special triangles" in geometry and trigonometry. They are the 30°-60°-90° right triangle that is half of an equilateral triangle, and the 45°-45°-90° isosceles right triangle that is half a square (cut by the diagonal).
The side ratios of these special triangles are relatively easy to remember. It is useful to memorize them.
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For the isosceles right triangle, the side lengths are the same. The Pythagorean theorem tells you that if they are both 1, then the hypotenuse is ...
√(1²+1²) = √2
That is, the side lengths of the 45-45-90 triangle are in the ratio ...
1 : 1 : √2
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For the triangle that is half an equilateral triangle, you know the hypotenuse is twice the length of the shortest side (since we got that short side by cutting a long side in half). Then the longer side can be found from the Pythagorean theorem:
√(2²-1²) = √3
That is, the side lengths of the 30-60-90 triangle are in the ratio ...
1 : √3 : 2
_____
In this problem, we're given the hypotenuse of a 30-60-90 triangle, so we know the short side of it (x) will be half that length:
x = (16√3)/2
x = 8√3
The hypotenuse of the 45-45-90 triangle will be √3 times x, so will be ...
long side of small triangle = (√3)(8√3) = 24
The shorter sides of that 45-45-90 triangle will be this value divided by the square root of 2, so are ...
y = z = 24/√2
We can multiply this by (√2)/(√2) to "rationalize the denominator".
y = z = 12√2
