the diagram below shoes a pennant in the shape of an isosceles triangle. the equal sides each measure 13, the altitude is x+7 and the base is 2x.

To find the value of x and the length of the altitude, we can use the properties of an isosceles triangle.

First, let's define the given information:
- Equal sides: 13 units each
- Altitude: x + 7 units
- Base: 2x units

Since the triangle is isosceles, the two equal sides are 13 units each. Let's call the other side, which is the base, 2x units.

Using the Pythagorean theorem, we can find the value of x.

In a right triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides.

Using this theorem, we have:
(2x)^2 = 13^2 - (x + 7)^2

Simplifying the equation:
4x^2 = 169 - (x + 7)^2

Expanding and simplifying further:
4x^2 = 169 - (x^2 + 14x + 49)

Distributing the negative sign:
4x^2 = 169 - x^2 - 14x - 49

Combining like terms:
4x^2 + x^2 + 14x + 49 - 169 = 0

5x^2 + 14x - 120 = 0

Now we can solve this quadratic equation for x either by factoring, completing the square, or using the quadratic formula.

Assuming the equation can be factored, we can rewrite it as:
(x + 12)(5x - 10) = 0

Setting each factor to zero:
x + 12 = 0 or 5x - 10 = 0

Solving for x:
x = -12 or x = 2

Since the length cannot be negative, we take x = 2 as our solution.

Therefore, the value of x is 2.

Now, we can find the length of the altitude using this value of x.

Given:
Altitude = x + 7

Substituting x = 2:
Altitude = 2 + 7 = 9 units

Therefore, the length of the altitude is 9 units.

To solve for the value of x, we can use the properties of an isosceles triangle.

First, we need to understand the relationship between the sides and the altitude in an isosceles triangle.

In an isosceles triangle, the two equal sides are called the legs, and the remaining side is called the base. The altitude is the perpendicular distance from the base to the opposite vertex.

In this case, the equal sides of the triangle measure 13, the altitude is x+7, and the base is 2x.

To find the value of x, we can use the Pythagorean theorem, which states that in a right triangle, the sum of the squares of the two legs is equal to the square of the hypotenuse.

In our case, the altitude (x+7) is the leg of a right triangle, and the base (2x) is the other leg. The hypotenuse of this right triangle is the equal side of the isosceles triangle, which measures 13.

Applying the Pythagorean theorem, we can set up the following equation:

(x+7)^2 + (2x)^2 = 13^2

Expanding and simplifying the equation gives:

x^2 + 14x + 49 + 4x^2 = 169

Combining like terms gives:

5x^2 + 14x + 49 = 169

Now, subtracting 169 from both sides of the equation:

5x^2 + 14x + 49 - 169 = 0

Rearranging the terms:

5x^2 + 14x - 120 = 0

Now, we can solve this quadratic equation by factoring, completing the square, or using the quadratic formula. The solutions for x will give us the value of x that satisfies the equation and corresponds to the given triangle.

Once we have the value of x, we can substitute it back into the expressions for the altitude (x+7) and the base (2x) to find their respective lengths.

Draw the figure

x^2 + (x+7)^2 = 13^2
x^2 + x^2 + 14x + 49 = 169
x^2 + 7x - 60 = 0
(x+12)(x-5) = 0

x = 5

sides are 13,13,10
altitude is 12, making the half-triangle a 5,12,13 triangle.