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一、填空題（每小题 3 分共 18 分） 1. 函数 $\mathrm{y}=\frac{\sqrt{3-2 \mathrm{x}}}{\mathrm{x}+...
Aug 15, 2024
Solution by Steps
step 1
Given that AC+BD=24 cm AC + BD = 24 \, \text{cm} and the perimeter of triangle OAB OAB is 18 cm 18 \, \text{cm} , we can express the lengths of OA OA , OB OB , AB AB as follows: OA+OB+AB=18 OA + OB + AB = 18
step 2
Since E E and F F are midpoints of AO AO and BO BO respectively, we have AE=12OA AE = \frac{1}{2} OA and BF=12OB BF = \frac{1}{2} OB . Thus, EF=12(AB) EF = \frac{1}{2}(AB)
step 3
To find AB AB , we can express AB AB in terms of OA OA and OB OB : AB=18−OA−OB AB = 18 - OA - OB . Substituting OA+OB=24−AB OA + OB = 24 - AB gives us AB+OA+OB=24 AB + OA + OB = 24
step 4
Solving the equations, we find OA+OB=6 OA + OB = 6 and AB=12 AB = 12 . Therefore, EF=12(12)=6 cm EF = \frac{1}{2}(12) = 6 \, \text{cm}
Answer
The length of EF EF is 6 cm 6 \, \text{cm} .
Key Concept
The properties of midpoints in a triangle and the relationship between the sides of a triangle and its perimeter.
Explanation
By using the properties of midpoints and the perimeter of triangle OAB OAB , we were able to derive the length of segment EF EF .
Generated Graph
Solution by Steps
step 1
Given that x < 2 , we simplify (x−2)2+∣3−x∣ \sqrt{(x-2)^{2}} + |3-x| . Since x < 2 , we have ∣3−x∣=3−x |3-x| = 3-x
step 2
The expression becomes (x−2)2+(3−x) \sqrt{(x-2)^{2}} + (3-x) . The square root simplifies to ∣x−2∣ |x-2| , which equals 2−x 2-x for x < 2
step 3
Therefore, the expression simplifies to (2−x)+(3−x)=5−2x (2-x) + (3-x) = 5 - 2x
Answer
The correct result is 5−2x 5 - 2x
Key Concept
Simplifying expressions involving absolute values and square roots
Explanation
The simplification process involves recognizing the conditions for absolute values and applying them correctly to arrive at the final expression.
What is the value of ∣3−x∣|3-x| when x < 2?
Solution by Steps
step 1
Given that x < 2 , we simplify (x−2)2+∣3−x∣ \sqrt{(x-2)^{2}} + |3-x| . Since x < 2 , we have ∣3−x∣=3−x |3-x| = 3-x
step 2
The expression becomes (x−2)2+(3−x) \sqrt{(x-2)^{2}} + (3-x) . The square root simplifies to ∣x−2∣ |x-2| , which equals 2−x 2-x for x < 2
step 3
Therefore, the expression simplifies to (2−x)+(3−x)=5−2x (2-x) + (3-x) = 5 - 2x
Answer
The correct result is 5−2x 5 - 2x
Key Concept
Absolute value and square root properties
Explanation
The expression simplifies by recognizing the conditions on x x and applying the definitions of absolute value and square roots.
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