Question

1. in quadrilateral love, what is the distance of point o from the origin?

a) \\( \sqrt { a + b } \\) c) \\( \sqrt { a ^ { 2 } + b ^ { 2 } } \\)
b) \\( \sqrt { a - b } \\) d) \\( \sqrt { a ^ { 2 } - b ^ { 2 } } \\)

1. what are the coordinates of point l in isosceles trapezoid live?

a) \\( ( d - c, e ) \\) c) \\( ( d - c, 0 ) \\)
b) \\( ( c - d, e ) \\) d) \\( ( e - c, d ) \\)

1. if give is a square, what is the distance between the points v and g?

a) a c) \\( 2 \sqrt { ( a + b ) } \\)
b) 2a d) \\( 2 \sqrt { ( a - b ) } \\)

Explanation:

Question 11

Step1: Use the distance formula

The distance formula between two points \((x_1,y_1)\) and \((x_2,y_2)\) is \(d=\sqrt{(x_2 - x_1)^2+(y_2 - y_1)^2}\). For the origin \((0,0)\) and point \(O(a,b)\), \(x_1 = 0,y_1=0,x_2=a,y_2 = b\).

Step2: Substitute into the formula

Substitute the values into the formula: \(d=\sqrt{(a - 0)^2+(b - 0)^2}=\sqrt{a^{2}+b^{2}}\)

Step1: Analyze the properties of an isosceles trapezoid

In an isosceles trapezoid, the non - parallel sides are equal in length and the bases are parallel. The \(y\) - coordinate of point \(L\) is the same as the \(y\) - coordinate of point \(I\) (since \(LI\parallel EV\)), so the \(y\) - coordinate of \(L\) is \(e\).
The length of \(EV=d\) and the length of \(LI = c\). The \(x\) - coordinate of \(L\) can be found by symmetry. If we consider the \(x\) - axis, the \(x\) - coordinate of \(L\) is \(d - c\) (because the trapezoid is symmetric about the vertical line passing through the mid - point of \(EV\) and \(LI\)).

Step1: Assume the side length of the square

Let the side length of the square \(GIVE\) be \(s\). If \(E\) is the origin \((0,0)\) and \(V=(a,b)\), in a square, if we assume the center of the square is at the origin (by symmetry, since it's a square). The distance from the center of the square to a vertex \(V(a,b)\) is related to the side length. But if we use the property of a square's diagonal. The distance from \(V\) to \(G\):
If we assume the center of the square is at the mid - point of \(VG\). Let the side length of the square be \(l\). Using the Pythagorean theorem for the square (if we consider the fact that for a square with vertices \((x_1,y_1)\) and \((x_2,y_2)\) and assuming symmetry about the axes). If \(E=(0,0)\) and \(V=(a,b)\), and \(G\) is the opposite vertex of \(V\) in the square. The distance \(VG\) is \(2\sqrt{a^{2}+b^{2}}\) (incorrect approach). But if we assume the square has side length \(a\) (by looking at the \(x\) - coordinate difference if we consider a square with vertices on the coordinate axes in a simple case). Let's use the property of a square's diagonal. If we assume the square has vertices \(E(0,0)\), \(V(a,0)\) (simplifying for calculation, since the general formula for the distance between two opposite vertices of a square with side length \(a\) is \(s\sqrt{2}\), but if we consider the coordinates. If \(V=(a,0)\) and \(G=(-a,0)\) (assuming a square centered at the origin with sides parallel to the axes), the distance \(VG=\vert a-(-a)\vert=2a\)

Answer:

C. \(\sqrt{a^{2}+b^{2}}\)

Question 12