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Solve for Diagonal Length and Angle in a Rectangle with Given Segment Lengths and Angle
Mathematics
Grade 9 (Junior High School)
Question Content
In the following rectangle, $EI = 2x + 6$, $FI = 6x - 6$ and $m\\angle EIH = 98^\\circ$. Find $EG$ and $m\\angle IGF$.
Correct Answer
$EG = 24$, $m\\angle IGF = 41^\\circ$
Detailed Solution Steps
1
Step 1: Use the property of rectangle diagonals. In a rectangle, diagonals are equal and bisect each other, so $EI = FI$. Set up the equation: $2x + 6 = 6x - 6$.
2
Step 2: Solve for $x$. Subtract $2x$ from both sides: $6 = 4x - 6$. Add 6 to both sides: $12 = 4x$, so $x = 3$.
3
Step 3: Calculate the length of $EI$. Substitute $x=3$ into $EI = 2x+6$: $EI=2(3)+6=12$. Since diagonals bisect each other, $EG = 2\\times EI = 2\\times12=24$.
4
Step 4: Analyze angles. $\\angle EIH$ and $\\angle FIG$ are vertical angles, so $m\\angle FIG = 98^\\circ$. Also, $FI = IG$ (from diagonal bisection and equality), so $\\triangle FIG$ is isosceles with $FI=IG$.
5
Step 5: Calculate $m\\angle IGF$. The sum of angles in a triangle is $180^\\circ$. Let $m\\angle IGF = m\\angle IFG = y$. Then $98 + 2y = 180$. Solve for $y$: $2y=82$, so $y=41^\\circ$.
Knowledge Points Involved
1
Properties of Rectangle Diagonals
In a rectangle, the two diagonals are equal in length and bisect each other, meaning each diagonal is split into two equal segments by their intersection point. This property is used to set up equations for segment lengths and identify isosceles triangles formed by the diagonals.
2
Vertical Angles Theorem
Vertical angles (opposite angles formed by intersecting lines) are congruent, meaning they have equal measures. This is used to find the measure of $\\angle FIG$ from the given $\\angle EIH$.
3
Isosceles Triangle Angle Properties
In an isosceles triangle, the angles opposite the equal sides are congruent. Combined with the triangle angle sum theorem (the sum of interior angles of a triangle is $180^\\circ$), this is used to calculate unknown base angles of the triangle.
4
Triangle Angle Sum Theorem
The sum of the measures of the three interior angles of any triangle is always $180^\\circ$. This is applied to solve for the unknown base angles of $\\triangle FIG$ after identifying the vertex angle.
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