if the altitude and the base of a triangle are each increased by 4 inches the area of the triangle will be increased by 42 square inches. if the altitude is increased by 3 inches and the base is decreased by 2 inches the area will be increased by 5 square inches. find the base and the altitude of the triangle.
let original base be b and height be h
original area = bh/2
case1:
new base = b+4
new height = h+4
new area = (b+4)(h+4)/2 = bh/2 + 42
bh + 4b + 4h + 16 = bh + 84
4b + 4h = 68
b+h = 17 #1
case2:
new base = b-2
new height = h+3
new area = (b-2)(h+3)/2 = bh/2 + 5
bh + 3b - 2h - 6 = bh + 10
3b - 2h = 16 #2
#1 x 2 ---->2b+2h= 34
add to #2
5b = 50
b = 10
a = 7
original base is 10 inches
original height is 7 inches
check:
original area = 35
case1:
area = 14(11)/2 = 77 , it increased by 42
case2:
area = (10)(8)/2 = 40 , it increased by 5
all is good!
To solve this problem, let's assume that the base of the triangle is represented by 'b' inches and the altitude is represented by 'h' inches.
According to the given information:
1) If the altitude and the base of the triangle are each increased by 4 inches, the area is increased by 42 square inches.
This can be expressed as:
Area1 = (1/2) * (b + 4) * (h + 4)
2) If the altitude is increased by 3 inches and the base is decreased by 2 inches, the area is increased by 5 square inches.
This can be expressed as:
Area2 = (1/2) * (b - 2) * (h + 3)
We can now set up a system of equations using these expressions:
Area1 - Area = 42 ....(Equation 1)
Area2 - Area = 5 ....(Equation 2)
Substituting the corresponding expressions for Area1 and Area2:
(1/2)(b + 4)(h + 4) - (1/2)bh = 42 ....(Equation 3)
(1/2)(b - 2)(h + 3) - (1/2)bh = 5 ....(Equation 4)
Now, we need to solve the system of equations (Equation 3 and Equation 4) to find the values of 'b' and 'h'.
Let's simplify and expand both equations:
((1/2)bh + 2h + 2b + 8) - (1/2)bh = 42 ....Expanding Equation 3
((1/2)bh - h - 3b - 6) - (1/2)bh = 5 ....Expanding Equation 4
Eliminating the (1/2)bh term:
2h + 2b + 8 = 42 ....Simplifying Equation 3
-h - 3b - 6 = 5 ....Simplifying Equation 4
Now, we have a system of two linear equations:
1) 2h + 2b = 42 - 8 ....Simplified Equation 3
2) -h - 3b = 5 + 6 ....Simplified Equation 4
Simplifying further:
1) h + b = 17 ....Equation 5 (Dividing Equation 3 by 2)
2) -h - 3b = 11 ....Equation 6 (Adding 1 to both sides of Equation 4)
Now, we can solve the system of equations (Equation 5 and Equation 6) to find the values of 'b' and 'h'.
Adding Equation 5 and Equation 6:
h + b - h - 3b = 17 + 11
-2b = 28
b = -14
Substituting the value of 'b' back into Equation 5:
h + (-14) = 17
h - 14 = 17
h = 31
Therefore, the base of the triangle is -14 inches and the altitude of the triangle is 31 inches.