6 Geometry questions?

Question

11.
The hypotenuse of an isosceles right triangle is 8 inches. Find the length of the legs.

16.
A tent is pitched such that it creates an equilateral triangle. If each of the congruent sides measure 2 meters, what is the height of the tent?

17.
An equilateral triangle has an altitude of 7 cm. Find the length of one side.

26.
What is the perimeter of a square with a diagonal of 3√2?

27.
What is the length of a diagonal of a square with a side of length 8?

28.
What is the length of the diagonal of a square with a 15-decimeter side?

Answer 1

Q.11.
The hypotenuse of an isosceles right triangle is 8 inches. Find the length of the legs.
Ans:
Since the triangle is an isosceles triangle therefore other two sides are equal. let each leg =x. Use pythegorean Theorm
x^2 +x^2 =(8)^2...............i
2x^2 =64 0r x^2=32 or x=sqrt 32=4*(sqrt 2)........ii
Q.16.
A tent is pitched such that it creates an equilateral triangle. If each of the congruent sides measure 2 meters, what is the height of the tent?
Ans: First find mid points of the sides, then join all midpoints with opposite vertex . Where they meet is centriod of a triangle.The distance of centriod from any vertex will be(2/3)√5 Using pythogorean therom
h^2 +{(2/3)√5 }^2=2^2
h^2 = -{(2/3)√5 }^2+2^2=-2.222222 +4=1.777777777....
h=4/3 units
Q.17.
An equilateral triangle has an altitude of 7 cm. Find the length of one side.
Ans:
Let each side of a triangle =2x, therfore half of it will be x.
Therefore use pythegorean Theorm
(x)^2 +(7)^2 = (2x)^2
on simplifying
3x^2=49 0r x= 4.04 units
Therfore side = 2x =8.083 units=8units app

Q.26.
What is the perimeter of a square with a diagonal of 3√2?
Ans:
If the side of the square is x then diagnol using pythegorean theorm is (x*sqrt 2)
x√2=3√2 or x=3
Perimeter = 4x =(4)*(3)=12 units.................
Q.27.
What is the length of a diagonal of a square with a side of length 8?
Ans:
As we know above that the length of diagnol =side of the square√2
Therfore dignol =8√2=11.3137085 units

Q.28.
What is the length of the diagonal of a square with a 15-decimeter side?
Ans:
Same as Q.27
Length of dignol =15√2decimeter=1.5(√2)=2.12132meter
or 21.21 decmeter

Answer 2

All these refer to right triangles and you use the Pythagorean Theorem: a^2 + b^2 =c^2, where a and b are the legs and c is the hypotenuse. Always draw a picture of your triangle.

11. Let x = both legs (isoceles triange), 8=hypotenuse
x^2 + x^2 = 8^2
2x^2 =64
x^2=32
x= square root of 32
either look it up, use your calculator or make it the square root of 16 times the square root of 2, or 4 times thesquare root of 2


16. Let the altitude (height)=a, b=1, c=2
a^2 +1^2=2^2
a^2 +1=4
a^2=3
a = square root of 3 which is 1.732

17. Let a=7, b=3.5, find c
49+12,25=c^2
16.25=c^2
c=4.03 (I used the calculator)

26. Let a and b = 8
64+64=c^2
128=c^2
c=11.314

28. Same problem, different number
Let a and b = 15dm
225+225=c^2
450=c^2
c=21.213

Answer 3

11. divide the hypotenuse by √2 (the square root of 2) for the length of each side. Answer: 5.6568542495 inches

16. Square the side of the tent (2²=4) then subtract the square of half of that length (4-1=3), then take the square root of that. Answer: √3 or 1.73205080757 meters

17. a=7/sin60°= 8.08290376866 centimeters

26. To get the length of one side of a square from the diagonal, divide the diagonal by √2. For the perimiter, multiply by 4. Answer: 12

27. Multiply the length of one side by √2. Answer: 8√2=11.313708499

28. Same as above 15√2=21.2132034356 decimeters.

Answer 4

1)According to Pythagoras theorem we have a^2 + b^2 = c^2 , whence a,b are legs and c is the hypotenuse.
Because the triangle is isosceles , 2a^2=c^2
Consequently, a= c/(squrt 2 )
In this case, a = 8/(squrt 2) =4 (squrt2)
2) Slant height at the center of base of the triangle is; squrt(2^2-1)=squrt 3.
Assuming base of the tent is a square, we have apothem of that square is 1
then height of the tent is ; squrt(1+3)=2 mts
3) Let a be the side of the triangle, then (a^2)/4 +7^2 = a^2
Also, a^2 - (a^2)/4 = 7^2
Solving this for a , we have, a = 7*(squrt3)
4) Diagonal of a square = (squrt 2)*( side of the square)
Hence, Side of the square = ( Diagonal of a square) / (squrt 2)
In this case, Side of the square =[ 3(squrt2)] /(squrt 2)
=3 ; hence, perimeter = 3*4 =12 units
5) Applying previous formula, we have,
Length of the diagonal of the square of side 8 units = 8(squrt 2)
6) Same as above.

Answer 5

(11) If length of a leg is L;
L^2 + L^2 = 8^2
L^2 = 32
L = 5.67 inches
.
(16) Height of tent = 2.Sin 60 = 2 (0.8660) = 1.732m
.
(17) Side x Sin 60 = 7cm
Side = 7/0.866 = 8.083 cm
.
(26) Diagonal^2 = side^2 + side^2
18 = 2. side^2
Side = 3
Perimeter = 12
.
(27) Diagonal ^2 = side^2 + side^2=64+64=128
Diagonal = 11.3137
.
(28) Diagonal = Sq. rt (15^2 + 15^2) = 21.213 decimeters

Answer 6

11.)
a^2 + b^2 = c^2
a = b
a^2 + a^2 = c^2
2a^2 = c^2

2a^2 = 8^2
2a^2 = 64
a^2 = 32
a = 4sqrt(2)

ANS : 4sqrt(2) inches

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16.)
h^2 + (a/2)^2 = a^2
h^2 + (2/2)^2 = 2^2
h^2 + 1^2 = 4
h^2 + 1 = 4
h^2 = 3
h = sqrt(3)

ANS : h = sqrt(3)

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17.)
h^2 + (a/2)^2 = a^2
7^2 + (a/2)^2 = a^2
a^2 - (a/2)^2 = 49
a^2 - ((a^2)/4) = 49
(4a^2 - a^2)/4 = 49
(3a^2)/4 = 49
3a^2 = 196
a^2 = (196/3)
a = 14(1/sqrt(3)) = (14sqrt(3))/3

ANS : (14/3)sqrt(3)

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26.)
a^2 + b^2 = c^2
a = b
a^2 + a^2 = c^2
2a^2 = c^2

3sqrt(2) = sqrt(18)

2a^2 = (sqrt(18))^2
2a^2 = 18
a^2 = 9
a = 3

3

4 * 3 = 12

ANS : 12

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27.)
2a^2 = c^2
2(8)^2 = c^2
2 * 64 = c^2
128 = c^2
c = 8sqrt(2)

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28.)
2(15)^2 = c^2
2(225) = c^2
c = 15sqrt(2)

ANS : 15sqrt(2) dm

Answer 7

11.
64 = 2x^2
x^2=32
x=sqrt(32)

17.
x^2 /4+ x^2=49
5/4 x^2=49
x^2=49(4)/5
x=14/sqrt(5)

Answer 8

11. a--->leg
2a^2=8^2
2a^2=64
a^2=32
a=5.656in

only knew that one....