In triangle ABC, CD is both the median and the altitude. If AB=5x+3, AC=2x+8, and BC=3x+5, what is the perimeter of triangle ABC?
Since CD is BOTH the median and altitude, the triangle is isosceles
and AB is the short side. The 2 long
sides(AB and BC) are equal.
AB = BC,
2X + 8 = 3X + 5,
2X - 3X = 5 - 8,
-X = -3,
X = 3.
P = AB + AC + BC,
P = (5X + 3) + (2X + 8) + (3X + 5),
P = (5*3 + 3) + (2*3 + 8) + (3*3 + 5),
P = 18 + 14 + 14 = 46.
fine the perimeter of ABCD = 46. AB = x+9, CD= 2x-7
1.05] Find the distance between the points (3, 0) and (-1, 5)
A, B, and C are collinear, AB = 5x - 19, and BC = 3x + 4. Find an
expression for AC if B is between A and C.
To find the perimeter of triangle ABC, we need to add up the lengths of all three sides.
Given that CD is both the median and the altitude, we can use the properties of a median and an altitude to find its length.
1. Median property: A median of a triangle divides the opposite side into two equal segments. In this case, CD divides AB into two segments of equal length. So, AD = DB.
2. Altitude property: An altitude of a triangle is the perpendicular distance from a vertex to the opposite side. In this case, CD is the altitude from vertex C to side AB. Since CD is perpendicular to AB, we have a right triangle ACD.
Now let's find the length of CD using the information given. Since AD = DB, we have AD = CD/2 and DB = CD/2.
Applying the Pythagorean theorem in triangle ACD:
(AC)^2 = (AD)^2 + (CD)^2
Substituting the given values:
(2x + 8)^2 = (CD/2)^2 + (CD)^2
Simplifying:
4x^2 + 32x + 64 = (CD)^2/4 + (CD)^2
Multiplying both sides by 4 to clear the fraction:
16x^2 + 128x + 256 = (CD)^2 + 4(CD)^2
Combining like terms:
16x^2 + 128x + 256 = 5(CD)^2
Rearranging the equation:
5(CD)^2 - (CD)^2 = 16x^2 + 128x + 256
4(CD)^2 = 16x^2 + 128x + 256
Dividing both sides by 4:
(CD)^2 = (16x^2 + 128x + 256)/4
Simplifying:
(CD)^2 = 4x^2 + 32x + 64
Taking the square root of both sides:
CD = √(4x^2 + 32x + 64)
Now that we have the value of CD, we can find the length of AB, AC, and BC.
AB = 5x + 3
AC = 2x + 8
BC = 3x + 5
To find the perimeter, add up the lengths of all three sides:
Perimeter = AB + AC + BC
Substituting the given expressions:
Perimeter = (5x + 3) + (2x + 8) + (3x + 5)
Combining like terms:
Perimeter = 5x + 2x + 3 + 8 + 3x + 5
Perimeter = 10x + 16x + 16
Perimeter = 26x + 16
So, the perimeter of triangle ABC is 26x + 16.