Question

properties of a triangle
lesson 5: isosceles triangle conjectures

1. practice in \\( \triangle a b c \\), suppose \\( a b=15 \mathrm{~cm}, b c=15 \mathrm{~cm}, a d=2 x-8 \mathrm{~cm} \\),

and \\( d c=4 x-20 \mathrm{~cm} \\). solve for \\( x \\).

1. practice in \\( \triangle a b c \\) above, suppose \\( m \angle a b d=\left(x^{2}-5\

ight)^{*} \\) and \\( m \angle c b d=4 x \\). solve for \\( x \\)

Explanation:

Question 5

Step1: Use the property of isosceles triangle

In an isosceles triangle \( \triangle ABC\) with \(AB = BC=15\) cm, the altitude \(BD\) bisects \(AC\) (by the property of isosceles triangle: the altitude from the vertex to the base of an isosceles triangle bisects the base). So \(AD = DC\).

Step2: Set up the equation

Since \(AD = 2x - 8\) and \(DC=4x - 20\), we have the equation \(2x-8=4x - 20\).

Step3: Solve the equation

Subtract \(2x\) from both sides: \(-8 = 2x-20\).
Add \(20\) to both sides: \(2x=20 - 8\).
\(2x=12\).
Divide both sides by \(2\): \(x = 6\).

Step1: Use the property of isosceles triangle

In an isosceles triangle \( \triangle ABC\) with \(AB = BC\), the altitude \(BD\) bisects \(\angle ABC\) (by the property of isosceles triangle: the altitude from the vertex to the base of an isosceles triangle bisects the vertex - angle). So \(m\angle ABD=m\angle CBD\).

Step2: Set up the equation

Since \(m\angle ABD=(x^{2}-5)^{\circ}\) and \(m\angle CBD = 4x^{\circ}\), we have the equation \(x^{2}-5=4x\).

Step3: Rearrange the equation

\(x^{2}-4x - 5=0\).

Step4: Factor the quadratic equation

We can factor \(x^{2}-4x - 5\) as \((x - 5)(x+1)=0\) (using the formula \(ax^{2}+bx + c=a(x - x_1)(x - x_2)\), where \(x_{1,2}=\frac{-b\pm\sqrt{b^{2}-4ac}}{2a}\), here \(a = 1\), \(b=-4\), \(c=-5\), \(\Delta=b^{2}-4ac=(-4)^{2}-4\times1\times(-5)=16 + 20=36\), \(x=\frac{4\pm6}{2}\)).

Step5: Solve for \(x\)

Set each factor equal to zero:
\(x - 5=0\) gives \(x = 5\);
\(x+1=0\) gives \(x=-1\).
Since an angle measure cannot be negative when considering the context of the problem (angle measures are non - negative), we discard \(x=-1\).

Answer:

\(x = 6\)

Question 6