The base of triangle is 2cm less than the altitude and the area of the triangle is 60cm². calculate the altitude.
b = a-2
Area = ba/2
Substitute a-2 for b in the second equation and solve for a.
To solve this problem, let's follow these steps:
Step 1: Understand the problem.
The problem states that the base of a triangle is 2 cm less than the altitude, and the area of the triangle is given.
Step 2: Identify the formulas related to the problem.
The formulas we need are:
1. Area of a triangle = (1/2) * base * altitude
2. Area of a triangle = 60 cm² (given)
Step 3: Replace the given values.
We're given that the area of the triangle is 60 cm². So we can replace the value of the area in the formula for the area of a triangle.
60 cm² = (1/2) * base * altitude
Step 4: Substitute the relationship between the base and altitude.
We're told that the base is 2 cm less than the altitude. We can rewrite this relationship as: base = altitude - 2.
Substituting this relationship into the equation, we get:
60 cm² = (1/2) * (altitude - 2) * altitude
Step 5: Simplify the equation.
To solve for the altitude, we'll simplify the equation by multiplying through by 2 to eliminate the fraction.
120 cm² = (altitude - 2) * altitude
Step 6: Expand and rearrange the equation.
Expand the equation by multiplying altitude by altitude and distributing the -2.
120 cm² = altitude² - 2altitude
Rearrange the equation to place all terms on one side:
altitude² - 2altitude - 120 cm² = 0
Step 7: Solve the quadratic equation.
The equation is now in the form of a quadratic equation. We can solve it by factoring, completing the square, or using the quadratic formula.
In this case, we'll use factoring. We need to find two numbers whose product is -120 and whose sum is -2. After considering different possibilities, we find that -12 and 10 satisfy these conditions.
altitude² - 12altitude + 10altitude - 120 cm² = 0
altitude(altitude - 12) + 10(altitude - 12) = 0
(altitude + 10)(altitude - 12) = 0
Setting each factor equal to zero gives us two possible values for the altitude:
altitude + 10 = 0 -> altitude = -10 (extraneous solution, not applicable in this context)
altitude - 12 = 0 -> altitude = 12
Step 8: Check the solution.
Since the altitude of a triangle cannot be negative, we discard the extraneous solution and conclude that the altitude of the triangle is 12 cm.